Internal residual stress calculating device, non-transitory computer-readable medium, and internal residual stress calculating method

ABSTRACT

An internal residual stress calculating device includes a prediction unit that predicts a temporal variation in deformation which is received by a medium having an image formed thereon from a correcting device correcting a deformation, and a calculation unit that calculates an internal residual stress of the medium having passed through the correcting device on the basis of a relational expression including an elasticity term and a term related to a plastic deformation and the temporal variation in deformation predicted by the prediction unit.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is based on and claims priority under 35 USC 119 fromJapanese Patent Application No. 2011-073411 filed Mar. 29, 2011.

BACKGROUND Technical Field

The present invention relates to an internal residual stress calculatingdevice, anon-transitory computer-readable medium storing a programthereof, and an internal residual stress calculating method.

SUMMARY

According to an aspect of the invention, there is provided an internalresidual stress calculating device including a prediction unit thatpredicts a temporal variation in deformation which is received by amedium having an image formed thereon from a correcting devicecorrecting a deformation, and a calculation unit that calculates aninternal residual stress of the medium having passed through thecorrecting device on the basis of a relational expression including anelasticity term and a term related to a plastic deformation and thetemporal variation in deformation predicted by the prediction unit.

BRIEF DESCRIPTION OF THE DRAWINGS

Exemplary embodiments of the present invention will be described indetail based on the following figures, wherein:

FIG. 1 is a block diagram illustrating the configuration of an imageforming apparatus including an internal residual stress calculatingdevice according to an exemplary embodiment of the invention;

FIG. 2 is a diagram illustrating an example of a correcting device;

FIG. 3 is a diagram illustrating an example of a deformation history inthe correcting device;

FIG. 4 is a diagram illustrating another example of the deformationhistory in the correcting device;

FIG. 5 is a functional block diagram illustrating the configuration ofthe internal residual stress calculating device according to theexemplary embodiment of the invention;

FIG. 6 is a diagram illustrating an example of a lookup table used bythe internal residual stress calculating device according to theexemplary embodiment of the invention;

FIG. 7 is a diagram illustrating an example where points inside a mediumare set by the internal residual stress calculating device according tothe exemplary embodiment of the invention;

FIGS. 8A and 8B are diagrams illustrating an example a residual stresscalculated by the internal residual stress calculating device accordingto the exemplary embodiment of the invention;

FIG. 9 is a flowchart illustrating the operation flow of the internalresidual stress calculating device according to the exemplary embodimentof the invention; and

FIG. 10 is a diagram illustrating an operation example of the internalresidual stress calculating device according to the exemplary embodimentof the invention.

DETAILED DESCRIPTION

An exemplary embodiment of the invention will be described withreference to the accompanying drawings. An internal residual stresscalculating device 1 according to this exemplary embodiment is used, forexample, in an image forming apparatus having a so-called decurler (acorrecting device correcting a deformation of a medium). As shown inFIG. 1, the image forming apparatus having the internal residual stresscalculating device 1 according to this exemplary embodiment includes amedium supplying unit 2 that stores media (for example, sheet-like mediasuch as sheets of paper) on which an image should be formed, a transportunit 3 that carries a medium from the medium supplying unit 2, an imageforming unit 4 that forms an image on the medium transported by thetransport unit 3 on the basis of image information input from theoutside, a correction unit 5 that is disposed in the transport unit 3and that corrects a deformation of the medium having an image formedthereon which is transported by the transport unit 3, and an apparatuscontrol unit 6 that controls all of the units, in addition to theinternal residual stress calculating device 1.

Here, the apparatus control unit 6 receives an image formationinstruction from the outside of the image forming apparatus and controlsthe transport unit 3 to transport a medium stored in the mediumsupplying unit 2 to the image forming unit 4. The apparatus control unit6 controls the image forming unit 4 to form an image corresponding tothe instruction on the medium. The correction unit 5 corrects thedeformation of the medium having an image formed thereon. The apparatuscontrol unit 6 outputs information for specifying the rigidity of amedium as an image forming destination stored in the medium supplyingunit 2. This information may be given in advance by a user and the like.

The internal residual stress calculating device 1 according to thisexemplary embodiment includes a control unit 11, a storage unit 12, aninput unit 13, and an output unit 14.

The control unit 11 includes a program control device such as a centralprocessing unit (CPU). The control unit 11 works in accordance with aprogram stored in the storage unit 12. Specifically, the control unit 11predicts a temporal variation in deformation which is received by themedium having an image formed thereon from the correction unit 5 as acorrecting device correcting a deformation. The control unit 11 performsa process of calculating a predicted value of an internal residualstress of the medium having passed through the correction unit 5 on thebasis of a relational expression including an elasticity term and a termrelated to a plastic deformation and the predicted temporal variation indeformation. The details of the process performed by the control unit 11will be described in detail later.

The storage unit 12 is a memory device or the like and stores a programperformed by the control unit 11. The program may be provided in a statewhere it is stored on a computer-readable recording medium such as aDVD-ROM (Digital Versatile Disc Read Only Memory) and may be copied tothe storage unit 12. The storage unit 12 also serves as a work memory ofthe control unit 11.

The input unit 13 receives information representing the type of a mediumused to form an image from the apparatus control unit 6 and outputs thereceived information to the control unit 11. The output unit 14 outputsinformation or instructions input from the control unit 11 to thecorrection unit 5.

Specifically, as shown in FIG. 2, the correction unit 5 includes atleast a pair of rollers 21 a and 21 b pressing each other. Thecorrection unit 5 forcibly deforms the medium by applying a pressingforce to the medium between the rollers 21 a and 21 b. The correctionunit 5 can change the pressing force, for example, by changing aninter-shaft distance between the rollers 21 a and 21 b. The correctionunit 5 as a decurler is widely known and thus will not be repeatedlydescribed in detail.

Since the correction unit 5 has the above-mentioned configuration, adeformation shown in FIG. 3 is temporally applied to the medium passingthrough the correction unit 5 in the thickness direction of the medium.In FIG. 3, the horizontal axis represents the elapsed time and thevertical axis represents the deformation of a sheet of paper (thedeformation in the thickness direction of the medium in the initialstate). In FIG. 3, plural graphs are drawn to correspond to thestiffnesses of media, where a medium is greatly deformed by the rollers21 a and 21 b at the time of passing through time t1 (X), thedeformation occurs on one surface of the medium just after the medium isdischarged from the rollers 21 a and 21 b (at time t2), and thedeformation varies to the other surface as time elapses. In anotherexample, the correction unit 5 may include three pairs of rollers and amedium is sequentially deformed three times (X, Y, and Z) by the pairsof rollers as shown in FIG. 4.

A process of predicting a deformation caused in a medium, which isperformed by the control unit 11 of the internal residual stresscalculating device 1, will be described below. The control unit 11functionally includes a prediction unit 31, a calculation unit 32, and acorrection amount calculating unit 33, as shown in FIG. 5.

The prediction unit 31 predicts a temporal variation (curvature history)of the deformation received by the medium from the correction unit 5depending on the stiffness of the medium. This prediction is performedby calculating a numerical expression corresponding to the graphs shownin FIGS. 3 and 4 or data listing coordinates (time, deformation) ofpoints in the graphs at plural times. In an example of this exemplaryembodiment, deformation values d0, d1, . . . , dn at times t0, t1, t2, .. . , tn (where t0=0, ti−1<ti, ti+1−ti=dt, tn=T) obtained by dividing aperiod of time from a certain time t=0 before the correction in thecorrection unit 5 to a time t=T after the correction by a predeterminedinterval dt are stored as a table (lookup table, which is hereinafterreferred to as LUT) in the storage unit 12 as shown in FIG. 6. The LUTmay be created for the stiffness of the medium and may be stored in thestorage unit 12. In this case, a table corresponding to the stiffness ofthe medium specified by the information output from the apparatuscontrol unit 6 is read out.

In another example of this exemplary embodiment, an approximationformula or a theoretical formula corresponding to the graphs shown inFIGS. 3 and 4 are determined in advance and the prediction unit 31calculates the deformation values d0, d1, . . . , dn at times t0, t1,t2, . . . , tn (where t0=0, ti−1<ti, ti+1−ti=dt, tn=T) obtained bydividing a period of time from a certain time t=0 before the correctionin the correction unit 5 to a time t=T after the correction by apredetermined interval dt on the basis of the approximation formula orthe theoretical formula.

In this case, a parameter corresponding to the stiffness of the mediummay be included in the approximation formula or the like and a valuecorresponding to the stiffness of the medium specified by theinformation output from the apparatus control unit 6 may be calculated.Alternatively, different approximation formulas for the stiffnesses ofthe media may be determined and the values may be calculated using theapproximation formulas corresponding to the stiffness specified by theinformation output from the apparatus control unit 6. Here, anapproximation formula or an theoretical formula can be created, forexample, using a structural analysis technique.

In another example, the prediction unit 31 may determine whether an LUTcorresponding to the stiffness of a medium specified by the informationoutput from the apparatus control unit 6 is stored in the storage unit12, may acquire the deformation values d0, d1, . . . , dn at times t0,t1, t2, . . . , tn (where t0=0, ti−1<ti, ti+1−ti=dt, tn=T) obtained bydividing a period of time from a certain time t=0 before the correctionin the correction unit 5 to a time t=T after the correction by apredetermined interval dt with reference to the LUT when it isdetermined that the LUT is stored, and may calculate the deformationvalues d0, d1, . . . , dn at times t0, t1, t2, . . . , tn (where t0=0,ti−1<ti, tn=T) using the approximation formula created by the structuralanalysis technique when it is determined that the LUT is not stored.

The calculation unit 32 calculates an internal residual stress of themedium passing through the correction unit 5 using the calculationresult of the prediction unit 31. Specifically, the calculation unit 32calculates a temporal variation in distortion ε at the points inside themedium from the curvature history information acquired by the predictionunit 31.

Specifically, as shown in FIG. 7, the calculation unit 32 takes N pointsp1, p2, . . . , pN in the thickness direction in a virtual cutting planeat a predetermined position X (the center in the medium transportingdirection or the like) in the medium transporting direction and acquiresthe temporal variation (εi(t), where i=1, 2, . . . , N) of thedistortion 6 at the points. Here, the temporal variation of thedistortion ε at the points with an interval Δt in a predetermined timerange (for example, from t=Ts to Te) is acquired using a predeterminedtime interval Δt. The end of the time range is a time after the partcorresponding to the position X related to the calculation on the mediumpasses through the correction unit 5.

Here, when the time range (from t=0 to t=T) or the time interval dt ofthe deformation history is different from the calculation range or thetime subtraction Δt of the temporal variation in stress calculatedherein, the deformation history information can be interpolated orextrapolated to acquire Δs/Δt. The points inside the medium shown inFIG. 7 should be set in the direction perpendicular to the mediumtransporting direction as well as in the thickness direction and shouldbe considered two-dimensionally. However, it is considered that thecorrection unit 5 uniformly applies the deformation in the directionperpendicular to the medium transporting direction and thus the pointsare one-dimensionally considered. When the calculation istwo-dimensionally performed, the one-dimensionally performed calculationhas only to be repeated by the number of points set in the directionperpendicular to the medium transporting direction (when plural pointsare set in the medium transporting direction and the medium is modeledthree-dimensionally, the calculation has only to be repeated by thenumber of points similarly). Accordingly, a one-dimensional calculationis assumed herein.

The calculation unit 32 calculates the predicted value of the internalresidual stress using Expression (1) expressing the relationship betweenthe stress and the distortion. In the expression, K represents theelastic modulus, μ represents the viscosity coefficient, and cryrepresents the yield stress. The viscosity coefficient μ can beexpressed by Expression (4) using an initial viscosity coefficient μ₀, adistortion-dependent coefficient A, and a viscous element distortionε_(vis). The elastic modulus K and the yield stress σ_(Y) areinformation specific to a target medium and are determined in advance.

$\begin{matrix}{{{{{If}\mspace{14mu} - \sigma_{Y}} \leq \sigma \leq {\sigma_{Y}\mspace{14mu}{is}\mspace{14mu}{satisfied}}},{\overset{.}{\varepsilon} = \frac{\overset{.}{\sigma}}{K}}}{{Otherwise},{\overset{.}{\varepsilon} = {\frac{\overset{.}{\sigma}}{K} + {\frac{\sigma - {\sigma_{Y}(\varepsilon)}}{\mu}\mspace{14mu}{is}\mspace{14mu}{{used}.}}}}}} & (1) \\{\mu = {\mu_{0} + {A \cdot \varepsilon_{vis}}}} & (4)\end{matrix}$

Expression (1) includes an elasticity term representing the elasticityand a term related to a plastic deformation. The elasticity term isexpressed by Expression (5) and the term related to the plasticdeformation is expressed by Expression (6)

$\begin{matrix}{\overset{.}{\varepsilon} = \frac{\overset{.}{\sigma}}{K}} & (5) \\\frac{\sigma - {\sigma_{Y}(\varepsilon)}}{\mu} & (6)\end{matrix}$

As shown in FIG. 7, the calculation unit 32 takes N points p1, p2, . . ., pN in the thickness direction in a virtual cutting plane at apredetermined position (the center in the medium transporting directionor the like) in the medium transporting direction and expresses thestress at the points at time n by Expression (7).σ_(i) ^(n)(i=1,2, . . . ,N)  (7)In FIG. 7, the distortion on the upper surface in the drawing in whichthe medium is stretched is defined as a stretching distortion (ε>0) andthe distortion on the lower surface in the drawing in which the mediumis shrunken is defined as a shrinking distortion (ε<0). A state where nodistortion is present is a state of ε=0. When Expression (1) is changedto the form of a time subtraction (interval Δt) with respect to thestress, Expression (8) representing the forward difference of the stressat the points is obtained.

$\begin{matrix}{\mspace{79mu}{{{{{If}\mspace{14mu} - \sigma_{Y}} \leq \sigma \leq {\sigma_{Y}\mspace{14mu}{is}\mspace{14mu}{satisfied}}},{\sigma_{i}^{n + 1} = {\sigma_{i}^{n} + {K\;{\Delta\varepsilon}}}}}{{Otherwise},{\sigma_{i}^{n + 1} = {{\sigma_{i}^{n} \cdot {\exp\left( {- \frac{\Delta\; t}{\tau_{i}}} \right)}} + {{\left( {1 - {\exp\left( {- \frac{\Delta\; t}{\tau_{i}}} \right)}} \right) \cdot \left( {{\mu_{i}\frac{\Delta\varepsilon}{\Delta\; t}} + \sigma_{Y_{i}}^{n}} \right)}\mspace{14mu}{is}\mspace{14mu}{{used}.}}}}}}} & (8)\end{matrix}$The calculation unit 32 in this exemplary embodiment calculates thepredicted value of the internal residual stress of the medium passingthrough the correction unit 5 using Expression (8) based on Expression(1). Expression (8) includes Expression (9) expressing the elasticityterm representing the elasticity and Expression (10) expressing the termrelated to the plastic deformation.

$\begin{matrix}{\sigma_{i}^{n} \cdot {\exp\left( {- \frac{\Delta\; t}{\tau_{i}}} \right)}} & (9) \\{\left( {1 - {\exp\left( {- \frac{\Delta\; t}{\tau_{i}}} \right)}} \right) \cdot \left( {{\mu_{i}\frac{\Delta\varepsilon}{\Delta\; t}} + \sigma_{Y_{i}}^{n}} \right)} & (10)\end{matrix}$

Here, τ represents a time constant expressed by Expression (11). εYrepresents the yield limit distortion expressed by Expression (12).

$\begin{matrix}{\tau = \frac{\mu}{K}} & (11) \\{\varepsilon_{Y} = \frac{\sigma_{Y}}{K}} & (12)\end{matrix}$

The calculation unit 32 substitutes the calculated temporal variation ofdistortion ε, that is, Δε/Δt, for Expression (8), sequentiallycalculates the stress, and acquires the stress (the stress remaininginside because the medium passes through the correction unit 5 at thistime) at time t=Te at the points shown in FIG. 7. That is, thecalculation unit 32 acquires the values of the internal residual stressat time t=Te at the points when N points p1, p2, . . . , pN in thethickness direction are taken at a point X in the medium transportingdirection through the use of this calculation and outputs the acquiredvalues as information representing the predicted values of the internalresidual stress.

The correction amount calculating unit 33 calculates information of thecorrection amount for reducing the internal residual stress of themedium passing through the correction unit 5 using the information ofthe predicted values of the internal residual stress calculated by thecalculation unit 32. Specifically, a stress at which the balancingmoment with the predicted values of the internal residual stress at thepoints in the thickness direction is “0” is calculated (FIG. 8A). Thecorrection amount calculating unit 33 creates and outputs informationfor controlling the correction amount such as the inter-shaft distanceof the rollers of the correction unit 5 on the basis of the distributionof the calculated predicted values of the internal residual stress.Widely-known methods can be used to create the information forcontrolling the correction amount corresponding to the information onthe stress distribution and thus will not be described in detail.

In this exemplary embodiment, a deformation may be present in advance ina medium (which is referred to as an initial deformation). In this case,the calculation unit 32 of the control unit 11 may calculate theinternal stress (the initial internal stress) at the points shown inFIG. 7 on the basis of the initial deformation, may add the calculatedvalues to the predicted values of the internal residual stress,calculated by the use of Expression (8), at the corresponding points tocorrect the predicted values of the internal residual stress, and mayoutput the resultant values as new predicted values of the internalresidual stress. In this case, the correction amount calculating unit 33calculates the stress at which the balancing moment is “0” with respectto the corrected predicted values of the internal residual stress at thepoints in the thickness direction (FIG. 8B). The initial deformation isdetected in advance by the use of a device (not shown) detecting adeformation, which is disposed in the transport unit 3. The control unit11 receives the information of the deformation detected by the deviceand calculates the initial internal stress using the information.

The internal residual stress calculating device 1 according to thisexemplary embodiment has the above-mentioned configuration and operatesas described below. That is, as shown in FIG. 9, the internal residualstress calculating device 1 receives the information specifying thestiffness of a medium output from the apparatus control unit 6. Then,the internal residual stress calculating device 1 first determineswhether the initial deformation is present (S1). It is determined thatthe initial deformation is present when the information of thedeformation is input from the device detecting the initial deformationand the input deformation is not “0”, and it is determined that theinitial deformation is not present otherwise.

When it is determined in step S1 that the initial deformation is present(YES), the internal residual stress calculating device 1 calculates thevalue of the initial internal stress based on the initial deformation(S2). The internal residual stress calculating device 1 determineswhether the LUT corresponding to the stiffness of the medium specifiedby the information output from the apparatus control unit 6 is stored(S3). When it is determined in step S1 that the initial deformation isnot present (NO), the internal residual stress calculating device 1performs the process of step S3.

When it is determined in step S3 that the LUT is stored (YES), theinternal residual stress calculating device 1 reads the deformationvalues d0, d1, . . . , dn (the deformation history) at times t0, t1, t2,. . . , tn (where t0=0, ti−1<ti, ti+1−ti=dt, tn=T) obtained by dividinga period of time from a certain time t=0 before the correction in thecorrection unit 5 to a time t=T after the correction by a predeterminedinterval dt with reference to the stored LUT (S4).

When it is determined in step S3 that the LUT is not stored (NO), theinternal residual stress calculating device calculates the deformationvalues d0, d1, . . . , dn (the deformation history) at times t0, t1, t2,. . . , tn (where t0=0, ti−1<ti, ti+1−ti=dt, tn=T) obtained by dividinga period of time from a certain time t=0 before the correction in thecorrection unit 5 to a time t=T after the correction by a predeterminedinterval dt by the use of a predetermined approximation formula or thelike in consideration of a structural analysis method or the like (S5).

In this way, when the values of the temporal variation in deformationd0, d1, . . . , dn are calculated, the internal residual stresscalculating device 1 calculates the temporal variation in distortion εat the points, which are set in the thickness direction as shown in FIG.7, inside the medium on the basis of the values of the temporalvariation (S6).

The internal residual stress calculating device 1 calculates thepredicted values of the internal residual stress using Expression (1)expressing the relationship between the stress and the distortion.

$\begin{matrix}{{{{{If}\mspace{14mu} - \sigma_{Y}} \leq \sigma \leq {\sigma_{Y}\mspace{14mu}{is}\mspace{14mu}{satisfied}}},{\overset{.}{\varepsilon} = \frac{\overset{.}{\sigma}}{K}}}{{Otherwise},{\overset{.}{\varepsilon} = {\frac{\overset{.}{\sigma}}{K} + {\frac{\sigma - {\sigma_{Y}(\varepsilon)}}{\mu}\mspace{14mu}{is}\mspace{14mu}{{used}.}}}}}} & (1)\end{matrix}$Specifically, when Expression (1) is changed to the form of a timesubtraction (interval Δt) with respect to the stress, Expression (8)representing the forward difference of the stress at the points isobtained.

$\begin{matrix}{\mspace{79mu}{{{{{If}\mspace{14mu} - \sigma_{Y}} \leq \sigma \leq {\sigma_{Y}\mspace{14mu}{is}\mspace{14mu}{satisfied}}},{\sigma_{i}^{n + 1} = {\sigma_{i}^{n} + {K\;{\Delta\varepsilon}}}}}{{Otherwise},{\sigma_{i}^{n + 1} = {{\sigma_{i}^{n} \cdot {\exp\left( {- \frac{\Delta\; t}{\tau_{i}}} \right)}} + {{\left( {1 - {\exp\left( {- \frac{\Delta\; t}{\tau_{i}}} \right)}} \right) \cdot \left( {{\mu_{i}\frac{\Delta\varepsilon}{\Delta\; t}} + \sigma_{Y_{i}}^{n}} \right)}\mspace{14mu}{is}\mspace{14mu}{{used}.}}}}}}} & (8)\end{matrix}$The internal residual stress calculating device 1 substitutes thecalculated temporal variation of the distortion ε, that is, Δε/Δt, forExpression (8) to sequentially calculate the stress and obtains thepredicted values of the stress (the stress remaining inside because themedium passes through the correction unit 5 at this time) at time t=Teat the points shown in FIG. 7 (S7).

Then, the internal residual stress calculating device 1 determineswhether the initial deformation is present (S8). This determination maybe performed in the same way as in step S1.

When it is determined in step S8 that the initial deformation is present(YES), the internal residual stress calculating device 1 adds the valuesof the initial internal stress at the points based on the initialdeformation calculated in step S2 to the predicted values of theinternal residual stress at the corresponding points acquired in step S7and sets the resultant values as new predicted values (S9). The internalresidual stress calculating device 1 calculates the information of thecorrection amount with which the internal residual stress of the mediumpassing through the correction unit 5 is reduced using the informationof the predicted values of the internal residual stress calculated instep S9 (S10).

When it is determined in step S8 that the initial deformation is notpresent (NO), the internal residual stress calculating device 1 performsthe process of step S10, that is, calculates the information of thecorrection amount with which the internal residual stress of the mediumpassing through the correction unit 5 is reduced using the informationof the internal residual stress calculated in step S7.

The internal residual stress calculating device 1 outputs theinformation of the correction amount to the correction unit 5. Thecorrection unit 5 adjusts the correction amount on the basis of theinput information of the correction amount and corrects the transportedmedium.

The expression used to calculate the internal residual stress used bythe calculation unit 32 in this exemplary embodiment is not limited toExpression (8) obtained by changing Expression (1) into the form ofsubtraction, but may be an expression including a term related to aplastic deformation and an elasticity term, such as Expression (2) orExpression (3).

$\begin{matrix}{\;{{{{{If}\mspace{14mu} - \sigma_{Y}} \leq \sigma \leq {\sigma_{Y}\mspace{14mu}{is}\mspace{14mu}{satisfied}}},{\sigma_{i}^{n + 1} = {\sigma_{i}^{n} + {K\;{\Delta\varepsilon}}}}}{{Otherwise},{\sigma_{i}^{n + 1} = {{K\;{\Delta\varepsilon}} + {{\sigma_{Y_{i}}^{n}\left( {\varepsilon,\overset{.}{\varepsilon}} \right)}\mspace{14mu}{is}\mspace{14mu}{{used}.}}}}}}} & (2) \\{{{{{If}\mspace{14mu} - \sigma_{Y}} \leq \sigma \leq {\sigma_{Y}\mspace{14mu}{is}\mspace{14mu}{satisfied}}},{\sigma_{i}^{n + 1} = {\sigma_{i}^{n} + {K\;{\Delta\varepsilon}}}}}{{Otherwise},\begin{matrix}{\sigma_{i}^{n + 1} = {\sigma_{i}^{n} + {\left( {K - \frac{\sigma_{Y_{i}}^{n}\left( {\varepsilon,\overset{.}{\varepsilon}} \right)}{\varepsilon_{Y_{i}}^{n}\left( {\varepsilon,\overset{.}{\varepsilon}} \right)}} \right){\Delta\varepsilon}}}} \\{= {\sigma_{i}^{n} + {{K^{\prime}\left( {\varepsilon,\overset{.}{\varepsilon}} \right)}{\Delta\varepsilon}\mspace{14mu}{is}\mspace{14mu}{{used}.}}}}\end{matrix}}} & (3)\end{matrix}$

Here, Expression (13) represents the yield stress at time n (a functionbased on the distortion and the temporal variation in distortion) and K′represents an equivalent elastic modulus in a plastically deformedstate.σ_(Y) _(I) ^(n)  (13)

In this exemplary embodiment, the calculation unit 32 may use anexpression set in advance, for example, depending on the type of amedium input from the apparatus control unit 6 out of Expressions (8),(2), and (3).

Specifically, Expression (2) is suitable for a medium such as a coatedsheet of paper in which the plastic deformation can be easily caused andExpression (3) takes consideration of the internal residual stress in amedium having a high water content. Accordingly, the calculation unit 32receives the information for specifying the type of a medium from theapparatus control unit 6, may perform the calculation using Expression(2) when the medium to be corrected is a medium such as a coated sheetof paper in which plastic deformation can be easily caused, may performthe calculation using Expression (3) when the medium to be corrected isa medium having a high water content, and may perform the calculationusing Expression (8) (Expression (1)) otherwise.

According to this exemplary embodiment, it is possible to predict adeformation (a stress corresponding thereto) close to the measuredvalue, as shown in FIG. 10, compared with the case where only theelasticity is considered. FIG. 10 shows the deformation corresponding tothe stress calculated by the calculation unit 32 in this exemplaryembodiment. Even when the curvature history is different, it can be seenthat the calculation unit 32 of this exemplary embodiment calculates thestress corresponding to the values close to the measured values,compared with the case where only the elasticity is considered.

The foregoing description of the exemplary embodiments of the presentinvention has been provided for the purposes of illustration anddescription. It is not intended to be exhaustive or to limit theinvention to the precise forms disclosed. Obviously, many modificationsand variations will be apparent to practitioners skilled in the art. Theembodiments were chosen and described in order to best explain theprinciples of the invention and its practical applications, therebyenabling others skilled in the art to understand the invention forvarious embodiments and with the various modifications as are suited tothe particular use contemplated. It is intended that the scope of theinvention be defined by the following claims and their equivalents.

What is claimed is:
 1. An image forming apparatus comprising: aprediction unit that predicts a temporal variation in deformation whichis received by a medium having an image formed thereon from a correctingdevice correcting a deformation; a calculation unit that calculates aninternal residual stress of the medium having passed through thecorrecting device on the basis of a relational expression including anelasticity term and a term related to a plastic deformation, and thetemporal variation in deformation predicted by the prediction unit, theelasticity term including a component corresponding to a distortion atpredetermined points in the medium; and a correction unit that correctsa position of the medium, by forcibly deforming the medium, based uponthe calculation by the calculation unit, the forcibly deforming of themedium occurring by applying a pressing force to the medium.
 2. Theimage forming apparatus according to claim 1, wherein the relationalexpression used in the calculation unit is selected depending on thetype of the medium from expressions: $\begin{matrix}{{{{{If}\mspace{14mu} - \sigma_{Y}} \leq \sigma \leq {\sigma_{Y}\mspace{14mu}{is}\mspace{14mu}{satisfied}}},{\overset{.}{\varepsilon} = \frac{\overset{.}{\sigma}}{K}}}{Otherwise},{\overset{.}{\varepsilon} = {\frac{\overset{.}{\sigma}}{K} + {\frac{\sigma - {\sigma_{Y}(\varepsilon)}}{\mu}\mspace{14mu}{is}\mspace{14mu}{used}}}},} & (1) \\{\;{{{{{If}\mspace{14mu} - \sigma_{Y}} \leq \sigma \leq {\sigma_{Y}\mspace{14mu}{is}\mspace{14mu}{satisfied}}},{\sigma_{i}^{n + 1} = {\sigma_{i}^{n} + {K\;{\Delta\varepsilon}}}}}{{Otherwise},{\sigma_{i}^{n + 1} = {{K\;{\Delta\varepsilon}} + {{\sigma_{Y_{i}}^{n}\left( {\varepsilon,\overset{.}{\varepsilon}} \right)}\mspace{14mu}{is}\mspace{14mu}{used}}}},{and}}}} & (2) \\{{{{{If}\mspace{14mu} - \sigma_{Y}} \leq \sigma \leq {\sigma_{Y}\mspace{14mu}{is}\mspace{14mu}{satisfied}}},{\sigma_{i}^{n + 1} = {\sigma_{i}^{n} + {K\;{\Delta\varepsilon}}}}}{{Otherwise},\begin{matrix}{\sigma_{i}^{n + 1} = {\sigma_{i}^{n} + {\left( {K - \frac{\sigma_{Y_{i}}^{n}\left( {\varepsilon,\overset{.}{\varepsilon}} \right)}{\varepsilon_{Y_{i}}^{n}\left( {\varepsilon,\overset{.}{\varepsilon}} \right)}} \right){\Delta\varepsilon}}}} \\{{= {\sigma_{i}^{n} + {{K^{\prime}\left( {\varepsilon,\overset{.}{\varepsilon}} \right)}{\Delta\varepsilon}\mspace{14mu}{is}\mspace{14mu}{used}}}},}\end{matrix}}} & (3)\end{matrix}$ where K represents an elastic modulus, μ represents aviscosity coefficient, ε represents a distortion, σ_(Y) represents ayield stress, and σ_(i) ^(n) represents a stress at point i at time n.3. The image forming apparatus according to claim 1, further comprising:a storage unit that stores information representing the temporalvariation in deformation which is received by the medium from thecorrecting device in advance, wherein the prediction unit predicts thetemporal variation in deformation which is received by the medium fromthe correcting device on the basis of the information stored in thestorage unit.
 4. The image forming apparatus according to claim 2,further comprising: a storage unit that stores information representingthe temporal variation in deformation which is received by the mediumfrom the correcting device in advance, wherein the prediction unitpredicts the temporal variation in deformation which is received by themedium from the correcting device on the basis of the information storedin the storage unit.
 5. The image forming apparatus according to claim2, wherein Expression 2 is selected when the medium is a coated sheet ofpaper.
 6. The image forming apparatus according to claim 2, whereinExpression 3 is selected when the medium is a medium having a watercontent above a predetermined level.
 7. A non-transitorycomputer-readable medium storing a program which causes a computer toserve as: a prediction unit that predicts a temporal variation indeformation which is received by a medium having an image formed thereonfrom a correcting device correcting a deformation; a calculation unitthat calculates an internal residual stress of the medium having passedthrough the correcting device on the basis of a relational expressionincluding an elasticity term and a term related to a plastic deformationand the temporal variation in deformation predicted by the predictionunit, the elasticity term including a component corresponding to adistortion at predetermined points in the medium; and correction twitthat corrects a position of the medium, by forcibly deforming themedium, based upon the calculation by the calculation unit, the forciblydeforming of the medium occurring by applying a pressing force to themedium.
 8. A correction method comprising: predicting, using aprediction unit, a temporal variation in deformation which is receivedby a medium having an image formed thereon from a correcting devicecorrecting a deformation; calculating, using a calculation unit, aninternal residual stress of the medium having passed through thecorrecting device on the basis of a relational expression including anelasticity term and a term related to a plastic deformation and thetemporal variation in deformation predicted by the prediction unit, theelasticity term including a component corresponding to a distortion atpredetermined points in the medium; and correcting a position of themedium, by forcibly deforming the medium based upon the calculation bythe calculation unit, the forcibly deforming of the medium occurring byapplying a pressing force to the medium.